θ * f * θ F (s) ξ s s+h a sin(ωt- φ)

Transcription

1 Perfrmace Imprvemet ad Limitati i Extremum Seekig Ctrl Mirlav Krtic Departmet f AMES Uiverity f Califria, Sa Dieg La Jlla, CA krtic@ucd.edu phe: fax: Submitted t Sytem & Ctrl Letter December, 998 Abtract We prpe the iclui f a dyamic cmpeatr i the extremum eekig algrithm which imprve the tability ad perfrmace prpertie f the methd. Thi cmpeatr i added t the itegratr ued fr adaptati t imprve the verall relative degree ad phae repe f the extremum eekig lp. The cmpeatr i ptetially mre eective i accutig fr the plat dyamic tha the fte ued phae hiftig f the demdulati igal. We preet a detailed aalyi f the extremum eekig ytem baed averagig. Thi aalyi prvide tw liear mdel, e fr trackig referece chage ad the ther fr eitivity t ie, which er iight it hw dieret parameter iuece the perfrmace. Thi aalyi i le cervative tha i previu cae ad allw the ue f fater adaptati fr imprved traiet. We exted the extremum eekig methd t prblem f trackig chage i the et pit which are mre geeral tha tep fucti. Thi wrk wa upprted i part by the Uited Techlgie Reearch Ceter, ad i part by grat frm ONR, AFOSR, ad NSF.

2 Itrducti Depite a large amut f reearch ert expeded durig the perid betwee the 94' ad 97' [, 4, 5, 6,,, 3, 4, 5], ad depite huge ucce i practice [3, 9], the methd f \extremum eekig" remai withut a rm theretical fudati. Tw available urvey extremum eekig ctrl, the e by Sterby [6] ad Secti 3.3 i Atrm ad Wittemark [], pit at a lack f tability guaratee ad clear deig guidelie. Pieerig wrk tability aalyi baed averagig i a extremum eekig ytem date back t Meerkv []. I [8] we preeted the rt tability aalyi fr a prblem with a geeral liear dyamical plat which ivlve the ue f bth averagig ad igular perturbati. Hwever, the cditi imped were retrictive: the plat had t be very fat (quai-tatic) albeit a cmpletely geeral -ae liear ytem ad the adaptati gai had t be mall. I thi reprt we preet a tighter aalyi which remve thee cditi, ad prpe dyamic cmpeati fr prvidig tability guaratee, fat trackig f chage i the peratig pit, ad meauremet ie rejecti. The reprt pe i Secti with a mtivati fr icrpratig dyamic cmpeati i the extremum eekig algrithm. Secti 3 give a ummary f me baic prpertie f liear time-peridic ytem that are ued thrughut the reprt. Secti 4 i the cre f the reprt where we derive average liearized mdel f the cled-lp ytem. I Secti 5 we give a extei f the extremum eekig algrithm t trackig f chage i the ukw ptimal peratig pit. I Secti 6 we dicu hw the theretical reult f the reprt iuece pible chice f deig parameter i the extremum eekig algrithm, ad i Secti 7 we give a cae tudy f the iuece f the deig parameter i the extremum eekig cmpeatr. Extremum Seekig Algrithm with Dyamic Cmpeati We tart with Figure which hw the implemetati f the extremum eekig algrithm. The algrithm i applied t a plat that ha a equilibrium map with a extremum. Withut l f geerality we aume that thi extremum i a miimum, i which cae we ue k >. I cae it i a maximum, the adaptati gai k wuld be egative. The plat i repreeted a a cacade cmbiati f liear dyamic ad a tatic liearity. Withi thi limited cla f ytem, the liear dyamic are allwed t appear bth at the iput ad at the utput. Bth f the liear blck, F i () ad F (), will be required t be table. The utput blck F () will be much mre dicult t hadle. It i crucial t te that all f the plat cmpet are allwed t be ukw. The ly dierece betwee the cheme i Figure ad that i [8] i that the ew cheme emply a cmpeatr C() i the adaptati law. Thi cmpeatr will be ued t imprve the tability prpertie f the extremum eekig cheme ad, i particular, remve the requiremet that the adaptati gai k be mall. The eect f thi cmpeatr will be imilar t addig derivative acti t a prprtial ctrller t imprve the dampig. Similarly, thi cmpeatr ca be regarded a a phae-lead cmpeatr which imprve the phae margi i a lp with a high relative degree. Oe limitati t the peed f adaptati will be imped by the preece f the meauremet ie iput. The peridic perturbati ued i the lp perfrm mdulati ad demdulati, ad their rle i t make the extremum f the equilibrium map, which i at ad therefre appear a a zer gai blck, appear, i a time average ee, a a gai prprtial t the ecd derivative at the extremum. The rle f the wahut lter =( + h) i t elimiate the bia t the DC cmpet f the equilibrium map. The tatic liear blck f() i aumed t have a miimum fr =. Frm the Taylr expai f() = f( ) + f ( )(? ) + f ( )(? ) + O? (? ) 3 ; () with the miimum aumpti f ( ) =, ad by drppig the term f rder three ad higher, we get f() = f( ) + f ( )(? ) : () By apprpriate rermalizati f the prblem, e ca abrb f ( )= it the itegratr gai k. Detig f( ) = f, we get f() = f + (? ) : (3)

3 θ * f * Plat F () i θ f( θ) F () y - k C() ξ +h a Im{ C (j ω)exp(j ωt)} a i(ωt- φ) Figure : Implemetati f the extremum eekig algrithm with cmpeati. I thi frmat, f ad ca be regarded a diturbace iput. Befre we prceed t ur detailed aalyi, thi i a gd place t make a cmmet abut the tyle f aalyi ad tati. Strictly peakig, we cat ay that exprei () ad () are equal becaue they dier by O? (? ) 3. Hwever, t keep the preetati imple, i the equel the equality ig will mea bth equal ad \apprximately equal." The apprximatee may be i term f eglected higher rder term, r i term f averagig apprximati (which are valid fr high!). All f the tatemet that we will be makig ca be iterpreted i term f rigru averagig ad lcal tability therem. The rea we d't purue thee iterpretati here a i [8] i eae f preetati. Ather cveti we will fllw fr eae f preetati i a cmbied ue f time ad frequecy dmai ymbl. I particular, a trafer fucti i frt f a bracketed time fucti, uch a G()[u(t)], mea a time dmai igal btaied a a utput f G() drive by u(t). 3 Lemma Mdulati I the equel, we will eed the fllwig tw lemma. The ymbl?t repreet expetially decayig term. Lemma 3. If the trafer fucti H() ha all f it ple with egative real part, the, fr ay real, H()[i(!t? )] = Im H(j!)e j(!t? ) +?t : (4) Prf. Thi i a tadard mdulati prperty f the Laplace trafrm. We give it prf becaue f it imprtace ad fr cveiece f the reader: h H()[i(!t? )] = H() Im e j(!t? ) i = j H() h e j(!t? )? e?j(!t? ) i = H() j L?? j! e?j? H() + j! ej = H(j!) j L?? j! e?j? H(?j!) + j! ej

4 +partial fracti due t H()g = H(j!)e j(!t? )? H(?j!)e?j(!t? ) +?t j = Im H(j!)e j(!t? ) +?t : (5) Lemma 3. If the trafer fucti G() ad H() have all f their ple with egative real part, the fllwig tatemet i true fr ay real ad ay uifrmly buded z(t): G()[(H()[i(!t? )])z(t)] = Im e j(!t?) H(j!)G( + j!)[z(t)] +?t : (6) Prf. The lemma i prved uig the fllwig traightfrward calculati: h G()[(H()[i(!t? )])z(t)] = G() Im H(j!)e j(!t?) z(t) +?ti (by Lemma 3.) = Im e?j L? fg()h(j!)z( + j!)g +?t = Im e j(!t?) H(j!)L? fg( + j!)z()g +?t (by deiti f Laplace trafrm) = Im e j(!t?) H(j!)G( + j!)[z(t)] +?t : (7) The fllwig eaily veriable tatemet i frmulated a a lemma fr cveiet referece. Lemma 3.3 Fr ay ratial fucti A() ad B(; ) the fllwig i true: 4 Average Liear Mdel Im e j(!t? ) A(j!) Im e j(!t?) B(; j!)[z(t)] = Re e j(?) A(?j!)B(; j!)[z(t)]? Re e j(!t??) A(j!)B(; j!)[z(t)] : (8) Suppe C(), F i (), ad F () are all aympttically table. We tart with the equati gverig the mdel i Figure : y = F () f + (? ) (9) = F i ()C() a i!t k? () = a i(!t? ) [y + ] ; () + h where () fllw frm applyig Lemma 3. with H() = C() ad =. Let u dete (t) = F i ()C() [a i!t] () ~ =? + (t) (3) ~y = y? F ()[f ] : (4) 3

5 Firt we derive a average mdel fr ~ ad the fr ~y. Legthy calculati baed the abve equati ad emplyig Lemma 3.{3.3 lead t the fllwig relatihip ~ + ka F i ()C() = + " ; Re e j + j! F i (j!)c(j!) + j! + h F ( + j!)[ ] ~ (5) where " = kaf i ()C() i(!t? ) hf ()[f ] + + a (F i ()C()[i!t]) + + h ~ i?ka F i ()C() + j! Re e j(!t?) F i (j!)c(j!) + j! + h F ( + j!)[~ ] : (6) I thi calculati we have eglected the expetially decayig term. We have reduced the ytem t a frm where the left ide f (5) i LTI, ad ly " i time varyig ad liear. Next, we determie the average mdel. Fr averagig t be applicable! mut be large relative t all ther parameter i the prblem. The average f (6) i: Avef"g = Ave kaf i ()C() i i(!t? ) ha (F i ()C()[i!t]) + h Thu we arrive at the fllwig prpiti. = O? ka 3 : (7) Prpiti 4. Fr the ytem i Figure, the average liearized mdel relatig ad ~ i ~() () = + L() ; (8) where L() = ka = e j + j! + j! + h F ( + j!)f i (j!)c(j!) +e?j? j!? j! + h F (? j!)f i (?j!)c(?j!) F i ()C() : (9) Thi ytem will be aympttically table if all the zer f the trafer fucti + L() have egative real part. Thi will be the cae if ad ly if a egative feedback ytem with the blck ad ka = e j + j! + j! + h F ( + j!)f i (j!)c(j!) +e?j? j!? j! + h F (? j!)f i (?j!)c(?j!) () F i ()C() () i aympttically table. Sice the trafer fucti () a itegratr with a pitive gai i pitive real, by the paivity therem (ee, fr example, [7]), the feedback lp will be table fr all k > prvided Nte that thee term are t ly additive but al multiplicative. The multiplicative term cat detry tability. Thi ca be ee uig a tadard Grwall lemma type argumet. 4

6 the blck () i trictly pitive real. While it i ituitively clear hw C(),, ad h ca be ued t make () trictly pitive real, it i hard t make thi electi ytematic. Fllwig the cgurati i Figure, e wuld w be tempted, baed ituiti fr LTI feedback ytem, t expect that the average trafer fucti frm the diturbace + F ()[f ] t the utput errr ~y be ~y() () + F ()f () = L()? + L() : () Thi i hwever t the cae at all. The liear time-varyig ature f the ytem prevet u frm uig e average ytem t btai ather. The crrect reult happe t be, hwever, a tated i the fllwig prpiti. Prpiti 4. Fr the ytem i Figure, the average liearized mdel relatig F ()[f ] + ad ~y i ~y() () + F ()f () = M()? + M() ; (3) where M() = ka = + h F () e?j F i (?j!)c(?j!) F i( + j!)c( + j!) + j! +e j F i (j!)c(j!) F i(? j!)c(? j!) : (4)? j! Thi i btaied by rt derivig the relatihip ~y =?ka =F () (F i ()C()[i!t]) F i()c() i(!t? ) + h [~y + F ()[f ] + ] + ; (5) where h = F ()? + ~ i + : (6) Ntig that ha a O( ~ +a ) average, ad uig Lemma 3.{ 3.3, we btai the liearized average ytem i the frm (3). The dierece betwee the lp trafer fucti L() ad M() (which i a LTI aalyi wuld be expected t be the ame) i trikig. Thi dierece give rie t dieret tability criteria fr the average ytem (3) ad (8). A urpriig a it may be, the dierece betwee the tw average ytem i t dicult t explai. The tw repreetati f the ame ytem i Figure, fr which we derived average mdel, are related by a time-varyig chage f crdiate (i fact, there are tw dieret chage f crdiate depedig whether we are expreig ~y i term f ~ r vice vera). Thu, e cat expect the average ytem t be equivalet. Example 4. Let u cider a ytem with C() = F i () = F () = ad = give i Figure. Thi ytem i the implet example f a extremum eekig prblem where the plat i jut a tatic map ad the eekig algrithm de't emply ay cmpeati r demdulatr phae hift. The ~ average mdel i ad the ~y average mdel i ~() () = ( + h + h +! ) 3 + (h + ka ) + (h +! + ka h) + ka! ; (7) ~y() () + f () =?ka 3 + (h + ka ) +! +! h : (8) 5

7 f * θ * θ f( θ) y - k ξ +h a iωt Figure : Example A Ruth tet hw that bth trafer fucti are aympttically table fr all k > (althugh, it huld be ted that the averagig apprximati i valid ly whe ka i mall relative t!). Hwever, a imple rt lcu aalyi hw dieret behavir betwee the tw trafer fucti. Fr mall h the ~ ytem will have a diple ear the imagiary axi, which mea that a pair f cled-lp ple will be lightly damped but their iuece the time repe will be mir. O the ther had, the ~y ytem will have tw lightly damped cled-lp ple which are t due t a diple. Thi happe fr large!, which mea that fr large! the ytem will be eitive t meauremet ie. A large value f ka will dampe thee ple but it will itrduce tw cled-lp ple ear the rigi (ad, a metied abve, fr large ka the averagig ceae t be valid). Thi example hw that pible beet f icreaig the adaptati gai t trackig chage i are accmpaied by icreaed eitivity t ie. 5 Geeralized Scheme fr Prblem with N-Step Chage i ad f Suppe a chage i the plat peratig cditi (characterized by either r f ) i t a abrupt tep chage but a mre gradual ramp chage. I that cae, we wuld apprpriately iclude thi ifrmati i the extremum eekig cmpeatr. Let L f (t)g =? () (9) L ff (t)g = f? f () ; (3) where? ();? f () are kw fucti, ad ; f are ukw ctat. The geeralized extremum eekig cheme i hw i Figure 3. Fr implemetati, the cmpeatr C i () ad C () huld be che that C i ()? () ad C ()=? f () are prper trafer fucti. With thi geeralized cheme we have L() = ka = e j C ( + j!)? f ( + j!) F ( + j!)f i (j!)c i (j!) +e?j C (? j!)? f (? j!) F (? j!)f i (?j!)c i (?j!) F i ()C i ()? () (3) 6

8 θ * Γ θ () f * Γ f () F () i θ f( θ) F () Plat y - kc i () Γ θ () ξ C () Γ f () a Im{ C (j ω)exp(j ωt)} i a i(ωt- φ) Figure 3: Extei f the extremum eekig algrithm t -tep chage i ad f. ad M() = ka = C ()? f () F ()? e?j F i (?j!)c i (?j!)f i ( + j!)c i ( + j!)? ( + j!) +e j F i (j!)c i (j!)f i (? j!)c i (? j!)? (? j!) : (3) The iclui f trafer fucti? () ad? f () it the extremum eekig lp achieve the cacellati f the ple f? () ad? f (), viz., ~() = L? () L() L f (t)g = + L? ()? () (33) ~y() =? M() + M() L ff (t)g =? f M() + M()? f () ; (34) that ~ (t) ad ~y(t) expetially decay t zer wheever the cmpeatr C i () ad C () are elected that all the zer f the trafer fucti + L() ad + M() have egative real part. Nte that me trackig bjective, uch a, fr example, a ramp i (t), may make the tabilizati prblem quite dicult (by itrducig a duble itegratr i? ()). 6 Deig Ciderati Let u w retur t the baic extremum eekig cheme give i Figure ad aalyzed i Secti 4. The eece f the reult i that, wheever pible, we huld che the cmpeatr C() t make the trafer fucti () trictly pitive real. Let u recall the eceary cditi fr SPRe: the trafer fucti mut be table, miimum phae, ad have relative degree larger tha e. Sice the adaptati law kc()= mut be prper (we cat ue pure dieretiati), the relative degree f C() mut be at leat -. The, fr () t have relative degree higher tha e, it mea that the relative degree f F ()F i () ca be at mt tw. Clearly, a cmpeatr C() f relative degree - will be typically f the prprtial-derivative (PD) type. The abve dicui utlie a typical ue f the cmpeatr C() i ituati where the verall relative degree f the plat ad the itegratr wuld exceed tw, i which cae high adaptati gai wuld 7

9 drive tw f the pe lp ple it the right half plae. The itrducti f C() wuld reduce the verall relative degree t tw ad allw the ue f higher adaptati gai k fr imprvig the cvergece. Belw, we dicu the diadvatage f exceively icreaig k, ad the uderlyig trade-. The cmpeatr i equally ueful whe the plat i f relative degree e. I that cae the lp gai withut C() i tw ad, uder high k, the extremum eekig lp becme lightly damped at bet (r eve utable). Thi eect ha bee berved i umeru imulati ad experimet by reearcher wrkig i thi area wh have regitered verht ad itabilitie. The iclui f a PD acti thrugh C() imprve the adaptati traiet ad tability prpertie. Hwever, there are ituati where PD acti via C() i ueceary ad may eve be harmful. Fr example, whe the plat already ha high badwidth (fr example, a lag-type plat with a fat ple), addig PD acti will icreae the badwidth ad make the ytem uecearily eitive t ie. Similar eect with repect t ie ca be expected i mre dicult plat (the with higher relative degree). Addig a PD acti via C() wuld reduce verht ad imprve tability fr reaable value f the adaptati gai, but fr exceive value it wuld make the adaptati ctamiated by ie. All the dicui thu far addree ly the iterplay betwee the cmpeatr, the plat, ad the adaptati gai. There i hwever al a delicate balace betwee the ize f the deig parameter ka, ad!. I rder fr the averagig methd t be applicable,! huld be large (i relative term) with repect t ka. Al, the preece f the O? ka 3 errr i the ~ -ytem idicate that we huld keep ka mderate. O the ther had, mall ka will lw dw the cvergece. It i al imprtat t ee that! ad t ly ka aect the peed f adaptati, becaue the peed f adaptati i prprtial t jf i (j!)c(j!)j. Typically, F i ()C() will have me rll- at high frequecie. Therefre, if! i elected t high t atify the cditi f the averagig methd, the cvergece will be lw. Thu, a preferable value f! i that which i jut uciet t eparate the time cale f the perturbati igal a i!t ad the plat with the extremum eekig dyamic. 7 A Cae Study I thi ecti we cider the prblem f extremum eekig fr a plat with the trafer fucti F i () = ; F () = ( + p)( + ) : (35) Thrughut thi ecti, p will have a xed value p = ad will vary betwee zer ad pitive value. Ule pecically metied, the ther quatitie will have mial value q = ka = =,! = 4, h = 4, =, ad C() =. 7. Iuece f parameter ka,!, ad h Firt we cider the cae where =, i which cae we ue cmpeati (C() = ) ad phae hiftig ( = ), ad illutrate ly the eect f chagig q,!, ad h. Figure 4 hw reult fr q = 3 ad M() q = 5. The tw Bde plt are fr the trafer fucti (left) ad? (right). Thee tw +L() +M() trafer fucti crrepd, repectively, t the iput-utput relatihip betwee ad, ~ ad ad ~y. Udereath the Bde plt we hw, repectively, the tep repe f ~ t the tep iput i the ptimal parameter, ad the impule repe f ~y t the ie (i term f frequecy ctet, a -impule i repreetative f white meauremet ie). We ee i Figure 4 that the icreae f q imprve the trackig f chage i but it al icreae the eitivity t ie. Figure 5 hw the eect f icreaig! fr frequecie! = 3 (lid) ad! = 5 (dahed). O the left we hw the eect f chage i the trackig errr, ad the right the eect f meauremet ie the utput. The gure hw a tred that higher! mthe ut the trackig repe but make the repe t ie mre cillatry. Figure 6 hw the eect f icreaig the cut- frequecy h f the high-pa lter =( + h) fr h = (lid) ad h = 5 (dahed). Eve thugh h cat be icreaed idicrimiately (i which cae the lter wuld apprach a pure dieretiatr), the gure hw that higher h imprve the repe. 8

10 Bde Diagram Bde Diagram Phae (deg); Magitude (db) Phae (deg); Magitude (db) 4 Frequecy (rad/ec) Frequecy (rad/ec) Step Repe Impule Repe Time (ec.) Time (ec.) Figure 4: Reult fr = ad q = 3 (lid) ad q = 5 (dahed). The plt the left crrepd t the trafer fucti ad the plt the right are fr the trafer fucti? M(). +L() +M() Frm the abve ciderati we pick q = ad! = h = 4 a gd value fr the plat with p = ad =, ad with cmpeatr, C() =. Nte that C() i t eeded becaue fr =, the verall extremum eekig lp i relative degree tw. 7. Cmpeati fr the relative degree tw plat Nw we icreae frm zer t a pitive value. Figure 7 hw reult fr tw dieret value f. The lid curve are fr = :, a cae where the cled-lp ytem i aympttically table. The dahed curve are fr the value = :6 fr which the cled-lp ytem ge utable. Beide the fact that the cled lp ge utable fr a larger umber f (with q =,! = h = 4, ad C() = ), e huld te that, frm the rt lcu e ca ee that the cled lp i utable eve fr arbitrarily mall umber f q (due t a pitive zer ad a ple ear the rigi i L() ad a pair f imagiary ple i M()). Figure 8 hw reult fr = :6 i the preece f a dyamic cmpeatr C() = + d with d = :8 (lid) ad d = :4 (dahed). We berve that the cmpeatr tabilize the plat ad, fr d = :4, achieve reaably gd trackig ad ie rejecti perfrmace. Fr larger value f d, t hw here, the perfrmace tart t deterirate. It huld be ted that the cmpeatr i f the PD frm, uch that the verall adaptati law C()= i a PI cmpeatr. The derivative acti i C() prvide a phae-lead eect eceary fr tabilizati ad perfrmace imprvemet. 9

11 Bde Diagram Bde Diagram Phae (deg); Magitude (db) Phae (deg); Magitude (db) 4 Frequecy (rad/ec) Frequecy (rad/ec) Step Repe Impule Repe Time (ec.) Time (ec.) Figure 5: Reult fr = ad! = 3 (lid) ad! = 5 (dahed). The plt the left crrepd t the trafer fucti ad the plt the right are fr the trafer fucti? M(). +L() +M() Figure 9 hw reult fr = :6 i the abece f a dyamic cmpeatr (C() = ) with = :3 (lid) ad = :5 (dahed). The dahed plt i thi gure are t very dieret frm the dahed plt i Figure 8 ad they hw that the phae hift i the demdulati igal ca, t me extet, help the tabilizati ad perfrmace imprvemet tak. Hwever, thi methd f cmpeati fr the plat dyamic i ridde with me ther dicultie, a example f which i the appearace f a lw earreace i the ie-t-utput trafer fucti, which give the lw ettlig i the impule repe (dahed i Figure 9). Referece [] K. J. Atrm ad B. Wittemark, Adaptive Ctrl, d editi, Readig, MA: Addi-Weley, 995. [] P. F. Blackma, \Extremum-eekig regulatr," i J. H. Wetctt, Ed., A Expiti f Adaptive Ctrl, New Yrk, NY: The Macmilla Cmpay, 96. [3] C. S. Drapper ad Y. T. Li, \Priciple f ptimalizig ctrl ytem ad a applicati t the iteral cmbuti egie," ASME, vl. 6, pp. {6, 95, al i R. Oldeburger, Ed., Optimal ad Self-Optimizig Ctrl, Bt, MA: The M.I.T. Pre, 966.

12 Bde Diagram Bde Diagram Phae (deg); Magitude (db) Phae (deg); Magitude (db) 4 Frequecy (rad/ec) Frequecy (rad/ec) Step Repe Impule Repe Time (ec.) Time (ec.) Figure 6: Reult fr = ad h = (lid) ad h = 5 (dahed). The plt the left crrepd t the trafer fucti ad the plt the right are fr the trafer fucti? M(). +L() +M() [4] A. L. Frey, W. B. Deem, ad R. J. Altpeter, \Stability ad ptimal gai i extremum-eekig adaptive ctrl f a ga furace," Prceedig f the Third IFAC Wrld Cgre, Ld, 48A, 966. [5] O. L. R. Jacb ad G. C. Sherig, \Deig f a igle-iput iuidal-perturbati extremum-ctrl ytem," Prceedig IEE, vl. 5, pp. -7, 968. [6] V. V. Kazakevich, \Extremum ctrl f bject with iertia ad f utable bject," Sviet Phyic, Dkl. 5, pp , 96. [7] H. K. Khalil, Nliear Sytem, d editi, Eglewd Cli, NJ: Pretice Hall, 996. [8] M. Krtic ad H. H. Wag, \Stability f extremum eekig feedback fr geeral liear dyamic ytem," accepted fr Autmatica, 998. [9] M. Leblac, \Sur l'electricati de chemi de fer au mye de curat alteratif de frequece elevee," Revue Geerale de l'electricite, 9. [] S. M. Meerkv, \Aympttic Methd fr ivetigatig quaitatiary tate i ctiuu ytem f autmatic ptimizati," Autmati ad Remte Ctrl,., pp , 967.

13 Bde Diagram Bde Diagram Phae (deg); Magitude (db) 5 5 Phae (deg); Magitude (db) 4 Frequecy (rad/ec) Frequecy (rad/ec) Step Repe Impule Repe Time (ec.) Time (ec.) Figure 7: Reult fr = : (lid) ad = :6 (dahed). The plt the left crrepd t the trafer fucti ad the plt the right are fr the trafer fucti? M(). +L() +M() [] S. M. Meerkv, \Aympttic methd fr ivetigatig a cla f frced tate i extremal ytem," Autmati ad Remte Ctrl,., pp. 96-9, 967. [] S. M. Meerkv, \Aympttic methd fr ivetigatig tability f ctiuu ytem f autmatic ptimizati ubjected t diturbace acti," (i Ruia) Avtmatika i Telemekhaika,., pp. 4-4, 968. [3] I. S. Mrav, \Methd f extremum ctrl," Autmati & Remte Ctrl, vl. 8, pp. 77-9, 957. [4] I. I. Otrvkii, \Extremum regulati," Autmati & Remte Ctrl, vl. 8, pp. 9-97, 957. [5] A. A. Pervzvakii, \Ctiuu extremum ctrl ytem i the preece f radm ie," Autmati & Remte Ctrl, vl., pp , 96. [6] J. Sterby, \Extremum ctrl ytem: A area fr adaptive ctrl?" Preprit f the Jit America Ctrl Cferece, Sa Fracic, CA, 98, WA-A.

14 Bde Diagram Bde Diagram Phae (deg); Magitude (db) 5 5 Phae (deg); Magitude (db) 4 Frequecy (rad/ec) Frequecy (rad/ec) Step Repe Impule Repe Time (ec.) Time (ec.) Figure 8: Reult fr = :6 ad d = :8 (lid) ad d = :4 (dahed). The plt the left crrepd t the trafer fucti ad the plt the right are fr the trafer fucti? M(). +L() +M() 3

15 Bde Diagram Bde Diagram Phae (deg); Magitude (db) 5 5 Phae (deg); Magitude (db) 5 5 Frequecy (rad/ec) Frequecy (rad/ec) Step Repe Impule Repe Time (ec.) Time (ec.) Figure 9: Reult fr = :6 ad = :3 (lid) ad = :5 (dahed). The plt the left crrepd t the trafer fucti ad the plt the right are fr the trafer fucti? M(). +L() +M() 4